Quadratic Functions and Models

Lesson 3.1

Nonlinear Data

When the points of the function are plotted, they do not lie in a straight line.

This graph contains points from a quadratic function of the form

Contrast to linear

2( )f x a x b x c

( )f x m x b

General Form Quadratic functions have the standard form

y = ax2 + bx + c a, b, and c are constants a ≠ 0 (why?)

Quadratic functions graph as a parabola

Axis of Symmetry Parabolas are symmetric

about a vertical axis For y = ax2 + bx + c the axis

of symmetry is at

Given y = 3x2 + 8x What is the axis of symmetry?

2

bx

a

Vertex of the Parabola The vertex is the “point” of the

parabola The minimum value Can also be a maximum

What is the x-value of thevertex?

How can we find the y-value?

2

bx

a

( )2

by f x f

a

Vertex of the Parabola Given f(x) = x2 + 2x – 8 What is the x-value of the vertex?

What is the y-value of the vertex?

The vertex is at (-1, -9)

21

2 2 1

bx

a

( 1) 1 2 9 9f

Vertex of the Parabola Given f(x) = x2 + 2x – 8

Graph shows vertex at (-1, -9)

Note calculator’s ability to find vertex (minimum or maximum)

Other Quadratic Forms

Standard formy = ax2 + bx + c

Vertex formy = a (x – h)2 + k Then (h,k) is the vertex

Given f(x) = x2 + 2x – 8 Change to vertex form Hint, use completing the square

Identifying Quadratic Functions

How can you determine which of the following is quadratic or not?

2( ) 4 5f x x

2( ) 7 3g x x x

2

2( )

3h x

x

What determines whether a parabola opens down or up?

What determines whether a parabola opens down or up?

Assignment

Lesson 3.1A Page 183 Exercises 1 – 77 EOO

Modeling Quadratic Data

Consider the following table … Is it linear or non linear? Graph the data on your calculator

Modeling Quadratic Data

The calculator can do quadratic regression to find a modeling function

While in the data matrix mode Press F5, then specify QuadReg

Fill in remaining parameters as specified here.

Fill in remaining parameters as specified here.

Modeling Quadratic Data

Note results

Formula

Graph

Assignment

Lesson 3.1B Page 185 Exercises 97 – 109 EOO